### An Adjusted Gray Map Technique for Constructing Large Four-Level Uniform Designs

ELSAWAH A M1,2,3, VISHWAKARMA G K4, MOHAMED H S5, FANG Kai-Tai1,2,6

1. 1. Department of Statistics and Data Science, Faculty of Science and Technology, Beijing Normal UniversityHong Kong Baptist University United International College, Zhuhai 519087, China;
2. Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College, Zhuhai 519087, China;
3. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt;
4. Department of Mathematics & Computing, Indian Institute of Technology Dhanbad, Dhanbad 826004, India;
5. College of Transportation and Civil Engineering, Fujian Agriculture and Forestry University, Fuzhou 350002, China;
6. The Key Lab of Random Complex Structures and Data Analysis, The Chinese Academy of Sciences, Beijing 100190, China
• Received:2021-05-07 Revised:2021-08-22 Online:2023-01-25 Published:2023-02-09
• Supported by:
Elsawah’s work was supported by the UIC Research Grants with No. of (R201912 and R202010); the Curriculum Development and Teaching Enhancement with No. of (UICR0400046-21CTL); the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, BNU-HKBU United International College with No. of (2022B1212010006); and Guangdong Higher Education Upgrading Plan (2021-2025) with No. of (UICR0400001-22).

ELSAWAH A M, VISHWAKARMA G K, MOHAMED H S, FANG Kai-Tai. An Adjusted Gray Map Technique for Constructing Large Four-Level Uniform Designs[J]. Journal of Systems Science and Complexity, 2023, 36(1): 433-456.

A uniform experimental design (UED) is an extremely used powerful and efficient methodology for designing experiments with high-dimensional inputs, limited resources and unknown underlying models. A UED enjoys the following two significant advantages: (i) It is a robust design, since it does not require to specify a model before experimenters conduct their experiments; and (ii) it provides uniformly scatter design points in the experimental domain, thus it gives a good representation of this domain with fewer experimental trials (runs). Many real-life experiments involve hundreds or thousands of active factors and thus large UEDs are needed. Constructing large UEDs using the existing techniques is an NP-hard problem, an extremely time-consuming heuristic search process and a satisfactory result is not guaranteed. This paper presents a new effective and easy technique, adjusted Gray map technique (AGMT), for constructing (nearly) UEDs with large numbers of four-level factors and runs by converting designs with $s$ two-level factors and $n$ runs to (nearly) UEDs with $2^{t-1}s$ four-level factors and $2^tn$ runs for any $t\geq0$ using two simple transformation functions. Theoretical justifications for the uniformity of the resulting four-level designs are given, which provide some necessary and/or sufficient conditions for obtaining (nearly) uniform four-level designs. The results show that the AGMT is much easier and better than the existing widely used techniques and it can be effectively used to simply generate new recommended large (nearly) UEDs with four-level factors.
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